Wait! What are we counting? On Ambiguity and Units

By Christopher Danielson, posted October
07, 2016 —

In my last post, I argued that ambiguity is an important source of
mathematical activity and that it is too often overlooked in
favor of certainty in math classrooms. Although certainty and right answers
have an important and unique role in mathematics, we would do well to offer ambiguity
to students as well.

My examples in the earlier post were of
geometry. In this post, I’ll introduce you to the possibilities afforded by
ambiguous number and counting tasks. Ambiguity in counting usually stems from
unstated units. Many situations allow for use of multiple units; a simple
example is eggs. At my local grocery store, I can buy eggs in a container
labeled 2 1/2 dozen. Why are 30 eggs labeled as 2 1/2 dozen, but 18 eggs are
labeled just as 18 eggs? Who knows? But it makes clear that counting eggs has two
important units: (1) eggs, and (2) dozens of eggs.

There is no ambiguity when buying eggs at
the grocery store because the units are clearly labeled. But I recently had
lunch with my son at a Japanese restaurant that had its menu displayed on
wooden slats on the wall. One of the menu items caught my eye.

In this case, there are two units—pot
stickers and dollars. Which number refers to which unit? How do you know? The
ambiguity of missing units gives us a chance to discuss. Before you read
further, stop the nearest person and ask him or her which number refers to
which unit. I’ll wait.

Now contrast that conversation you just had
with what likely follows from this unambiguous textbook-like treatment of the
pot sticker situation:

A Japanese restaurant sells pot stickers 4
for $4, 8 for $7 or 12 for $9. What is the price per pot sticker for each of
these servings?

The pot sticker situation is one with a
right answer—one of those numbers really does refer to dollars, and the other
really does refer to pot stickers. Next, let’s consider a couple of situations
that have multiple right answers. Look at the picture below and consider the
question, How many?

Did you say two? Then you were probably
counting shoes. Did you say one? Maybe you were counting boxes or pairs of
shoes. Other common answers are four and twenty. What are the people with those
answers counting? “How many?” is an ambiguous question because it doesn’t
specify what you should count.

In most elementary school classrooms that I
have visited, asking this question quickly develops into a wonderful game in
which students find new things to count, challenge themselves to count more
difficult items, and even assign informal probabilities to their counts.
(“There are probably about fifty yellow stitches, but there could be more—or
less—because I can’t see them all.”)

Another fun “How Many?” prompt is about the
following photograph.

One, seven, eight, fifteen, and seven-and-a-half
are all common answers here. Each of them is correct as long as it is attached
to the right unit; and, of course, there are probably more possibilities that
your students will see that I haven’t anticipated.

The ambiguity of the question, “How many?”
with unspecified units can lead to discussions of whether a half-avocado counts
as a unit (answer: yes) and to the question of whether there is one-to-one
correspondence of pits to avocados (answer: not in this photo). These are deep,
important, and, above all, mathematical questions that come to the surface
because the task allows children the space to think and to be creative.

Your
turn

I hope you’ll share
these tasks with your students and colleagues, and I especially hope you’ll
share the results of doing so—either here in the comments or on Twitter (I’m
@trianglemancsd). What new possibilities do you see opening up as you and your
students wrestle with unstated units and ambiguous counting in math class?

Christopher Danielson is on the
teaching faculty at Desmos, which offers a set of free digital math tools along with a growing library of activities
developed by the community of users. He is the author of two books—Common
Core Math for Parents For Dummies, and Which One Doesn't Belong? A
Shapes Book. You can find more of his writing at his website: Talking Math with Your Kids. For full-size, classroom-ready versions of the images used in this post,
go to http://talkingmathwithkids.com/2016/10/10/images-for-counting-and-units/.